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Properties

Label 2-38e2-76.35-c0-0-2
Degree $2$
Conductor $1444$
Sign $0.688 - 0.725i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.107 − 0.608i)5-s + (−0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.473 − 0.397i)10-s + (1.52 + 0.553i)13-s + (−0.939 + 0.342i)16-s + (−1.23 − 1.04i)17-s + 18-s + 0.618·20-s + (0.580 + 0.211i)25-s + (0.809 + 1.40i)26-s + (−1.23 + 1.04i)29-s + (−0.939 − 0.342i)32-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.107 − 0.608i)5-s + (−0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.473 − 0.397i)10-s + (1.52 + 0.553i)13-s + (−0.939 + 0.342i)16-s + (−1.23 − 1.04i)17-s + 18-s + 0.618·20-s + (0.580 + 0.211i)25-s + (0.809 + 1.40i)26-s + (−1.23 + 1.04i)29-s + (−0.939 − 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.688 - 0.725i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.688 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.766717993\)
\(L(\frac12)\) \(\approx\) \(1.766717993\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.369416660430737557990219026653, −9.044363630681117621348722085522, −8.198552660583366363345076980762, −7.09071946008934206750068137333, −6.62031952670918300358629591739, −5.71563429866797069386597634243, −4.73067209624907666524288404488, −4.09044719084616686544817523125, −3.12372278887927279642290232269, −1.58359481563180506968529824179, 1.53716816421469704562920067556, 2.51019015334262736459366032698, 3.69976078942912118248525949074, 4.29221179272649407116716839494, 5.44303263137808453553548515356, 6.25881352312130541146679623037, 6.88905473290518352461044704287, 8.030400807294885984331097422626, 8.926503407394556420932941702371, 9.960543121644665481738832655604

Graph of the $Z$-function along the critical line